Entropy, sets

Hausdorff measures old and new, and limit of geometrically finite Kleinian groups

by

D E N N I S S U L L I V A N

I.H.E.S., Bures-sur- Yvette, France and

C.U.N,Y., Nero York, U.S.A.

Given a (closed) set A contained in the plane and a positive function ~p(r) (for example

r ~, r6(log I/r) 6', etc.) one defines the (covering) Hausdorff measure of A relative to ~(r)

by considering coverings of a subset A c A by balls Bi, B2 .... of radii rl, r2 .... all less

than e~>0. The (covering) Hausdorff ~o-measure of A is the limit as e---~0 of the infimum

over such coverings of the sums EiVd(ri).

We have been led by the study of limit sets of Kleinian groups to a dual construc-

tion based on packings rather than Coverings. Here one considers for U open in A

collections of disjoint(l) balls BI,B2 . . . . contained in U of radii rl,r2 . . . . all less than

e~>0. Then the packing Hausdorff ~p-measure of U is by definition the limit as e--*0 of

the supremum over such packings of the sums Ei~0(ri). If two open sets are disjoint

these limits add. It follows (w 7) that the outer measure defined by this function of open

sets defines a countably additive Borel measure.

The covering and packing Hausdorff measures can be distinguished rather dramati-

cally for the limit sets of Kleinian groups which are geometrically finite but have cusps.

For example the "Apollonian packing" limit set (Fig. 1) has for a certain 6 a

positive (locally finite) r 6 Hausdorf f measure for the covering definition (6-1.3). The

r ~ packing measure is locally infinite.

For a second example consider the packing obtained by the rectangular array of

unit diameter circles (Fig. 2). Inverting these in the dotted circle of unit diameter packs

them into one quadrilateral interstice. Now translate in both directions and repeat the

procedure ad infinitum to get the second example. This limit set has locally finite r 6

(~) Balls are disjoint if the dis tance between their centers is greater than the sum of the radii.

2 6 0 D. SULLIVAN

Fig. 1

Hausdorff measures with the packing definition, for some 1<6<2. The r a covering

measure is zero.

These examples are limit sets of subgroups of the Picard group, consisting of 2x2

matrices of Gaussian integers.

More generally, let F be a discrete subgroup of linear fractional transformations

{z--~(az+b)/(cz+d)} which is geometrically finite, that is F has a finite sided fundamen-

tal domain for its action on hyperbolic 3-space (thought of as the upper half space with

boundary C U ~ or the unit 3-ball with boundary the sphere). The limit set of F denoted

A=A(F), is the set of cluster points in C U ~ of any orbit ofF. In this paper we study the

geometry of A using the ergodic nature of the action of F on A.

If n(R) is the cardinality of the intersection of a fixed orbit of F in hyperbolic 3-

space with balls of hyperbolic radius R with fixed center, the critical exponent of F is

by definition the exponential growth rate of n(R),

6 = 6 (F) = lim I log n(R). R--,= R

The number 6(F) is the critical exponent for absolute convergence of Poincar6 series

associated to F (see w I).

Now let D=D(F) denote the Hausdorff dimension of the limit set A(F). Thus by

definition, D=infimum of the set of a so that the (covering) Hausdorff r a measure of

A(F) is zero (=supremum of the set of a where the covering Hausdorff r a measure of

A(F) is infinity).

Let [7'1 denote the linear distortion of any linear fractional transformation V in a

fixed spherical metric on C U ~.

We say that a finite measure It on the limit set is geometric if

?*/~= ly'l~ yEF.

(Here 7*/~(A) =/~(TA). )

THEOREM 1. For a geometrically finite group F, the critical exponent 6(I') equals

the Hausdorff dimension D(F) of the limit set A(F); and D(F)<2 /f A(F) is a proper

ENTROPY, HAUSDORFF MEASURES AND KLEINIAN GROUPS 261

subset o f C U oo. Moreover, there exists one and only one geometric measure It o f total

mass one supported on A(F).

We have tried for some time to describe the canonical geometric measure in terms

of the metric structure of the limit set alone (without using the group of conformal

symmetries A). The cusps present interesting difficulties.

B y a cusp o f F we mean a conjugacy class in F of maximal parabolic subgroups

fixing points of C U ~ (necessarily in A(F)). Such a subgroup either contains Z or Z + Z

of finite index and these are called rank one cusps or rank two cusps accordingly.

Le t It denote the canonical geometric measure (Theorem 1) on the limit set of a

geometrically finite group F. Le t vp and vc denote the packing and covering Hausdorf f

measures on A(F) using the gauge function r ~ (These measures may a priori be either

zero or not even or-finite.)

THEOREM 2. Unless there are cusps o f rank 1 and o f rank 2, the canonical

geometr ic measure it equals one or both o f the metrically def ined measures Vp or Vc.

The situation is described by the table:

Only rank 1 Only rank 2 Cusps of both No cusps cusps cusps ranks

D=2 /t=vp=vc Not possible /~=Vp=Vc Not possible =spherical measure

/~=?

1 < D < 2 /Z=Vp=Vc /~=Vc.Vp= oo i~=vp~:vc=c=O 0=Vc:~Vp=

D= 1 p=Vp=Vr /~=vp=v~ Not possible Not possible

D< I p=vp=vc p=Vp*V~=0 Not possible Not possible

Remark . (1) An interesting case is finitely generated Fuchsian groups of the second

kind with cusps (A(F) is a Cantor subset of a round circle contained in the Riemann

sphere). Looking at the table ( I / 2 < D < l , only rank one cusps) shows the canonical

geometric measure is the packing measure, but the covering measure is zero.

(2) A deeper analysis ([SI]) shows that if It is not equal to vp (or vc) then It is not

equivalent to a packing (or covering) measure relative to any gauge function. Thus limit

sets provide natural examples where packing measures are finite and positive but no

covering measure is (for any gauge function) and vice versa.

The canonical geometr ic measure It determines a F invariant measure It x i t / I x -y l 2D

f

D. SULLIVAN 262

Fig. 2

on pairs of points on the sphere. (2) Combining this measure with arc length determines

an invariant measure dm~ for the geodesic flow associated to H3/r '. This measure plays

a key role in the proofs of Theorems I and 2 using part (i) of

THEOREM 3. (i) The measure dm~, has finite total mass and is ergodic for the

geodesic flow. Moreover,

(ii) The measure theoretical entropy o f the geodesic flow relative to dm~ is D, the

Hausdorffdimension of A(F).

(iii) In case there are no cusps the geodesics with both endpoints in A(F)form a

compact invariant set, and the topological entropy is again D. Thus in this case dm~ is

a measure that maximizes entropy.

In summary we have defined the same real number in terms of the Poincar6 series,

the Hausdorff dimension of the limit set, and the entropy of the geodesic flow.

Historical note and acknowledgements. Most of Theorem I was proven for most

Fuchsian groups by Patterson in [P]. In particular, he constructed measures satisfying

y*p=ly'[ap for any discrete group and estimated an associated eigenfunction q0, (cf.

w Rufus Bowen [B] constructed such measures using Markov partitions and showed

D > 1 for quasi-Fuchsian compact surface groups.

(2) This follows from ~ ,p=ly , [og and lyx-yy[Z=[y'x[ [y'y[ Ix-Y[ 2 where [a-b[ is the Euclidean dis tance between a and b on the unit sphere.

ENTROPY, HAUSDORFF MEASURES AND KLEINIAN GROUPS 263

nk one cusp

Rank tw~ c u s p ~ e ~fo 1 ~

Boundary

Fig. 3

References of these two papers include those to earlier work by Akaza and A. F.

Beardon who also related 6(F) to the Hausdorff dimension of A(F), for certain Fuchsian

groups.

In [$2] we continued the development of these papers to study general properties

of such measures, 6 and D. We were able there to treat general Fuchsian groups (w 6)

and higher dimensional geometrically finite groups without cusps (w 3). Discussions of

that paper with Thurston (with crucial assists) allowed simplifications and completions

of the discussion given here for the general geometrically finite case with cusps. We

wrote the paper for dimension 3 but it clearly translates into dimension n.

Finally, in Thurston's hyperbolic geometry notes [T] one finds in revealing form

the geometric information about geometrically finite groups required here. Namely the

quotient by F of the convex hull of the limit set consists of a compact piece with

boundary and exponentially skinny ends attached, one for each cusp (Fig. 3).

w I. Existence of the (geometric) measure/l

For any discrete group of hyperbolic isometries define the absolute PoincarO series by

gs(x, y) = ~ exp ( - s . hyperbolic distance (x, ~y)), ~EF

264 D. S U L L I V A N

f I I T I I

I I I ; ) I/ /

t / //---~ T v> •

/ I , V

F i g . 4

and the critical exponent 6(F) as the infimum of the set of s where gs(x, y ) < ~ . Here x

and y are points in hyperbolic space, H 3.

Following a construction of Patterson [P] one places atomic masses along the orbit

{yy} of y with weights

1 - - exp ( - s. distance (x, yy)). gs(Y, Y)

The limit as s---~6(F) defines a measure/~ on the limit set A(F) satisfying

~'*~ = Jy'l%, ~'~ r , 6 = ~(r).

(See [$2] w 1. Actually if g a ( y , y ) < ~ one has to increase each term in the sum for

gs(x, y) by a factor h (distance (x, 7Y)) of arbitrarily small exponential growth to make

the series diverge at s=6 . For e>0, k>0, h(x) satisfies Ih(s+t)/h(s)-ll<e for O<.t<.k and

s sufficiently large.)

We will see later w 4 that 6(F) is the Hausdorff dimension of A(F) in the geometri-

cally finite case, and the Poincar6 series diverges at s=6. In particular the h-factor is

not necessary (after the fact).

w 2. /~ h a s n o a t o m s

We work in the upper half space model and we put a cusp at ~ . If the cusp has rank

one, we may assume A(F) is contained between two parallel lines y=0, y = l and F

contains the translation T(x+iy)=(x+iy)+l (See [T], p. 8.21) (Fig. 4).

ENTROPY, HAUSDORFF MEASURES AND KLEINIAN GROUPS 265

X//I Fig. 5

X p

r log z

We est imate gs(X', y)/gs(x,y) in terms of z, where x is a fixed point and x ' is at

height z above x ' , as s approaches 6(F). Each sum over F is broken into a sum 2 '

corresponding to the part of the orbit of y clustering down to a fundamental domain of T

and then a sum over the integers Z corresponding to {Tn)nez.

The E' sums for x ' and x are in the approx imate ratio z -~ for s near 6.

The entire sum for x is obtained by some (fixed) convergent series (3) multiplying

the Z' term. The same relation holds for x ' . The convergent series for x' is comparab le

to the integral

1 ~ I 6 f r - 2~ d t = z ~ (~+t2) dt.

1 + (t/z) /

Since the integral is convergent , 6>1/2. More generally a cusp of rank k implies

6(F)>k/2 (Fig. 5) (cf. [B2]).

The combined effect gives

gs(x', y)/gs(X, y) ~ z I -~

(ignoring the effect o f the factors h (distance (x, 7'Y)) which are smaller than any power

of z for z large).

I f we had put a rank 2 cusp at infinity the argument would involve a sum over a

lattice (or a double integral) and the factor z 2-~ emerges.

I f kt has an a tom at the cusp the ratio would increase as fast as z ~ as x'--->~

(3) The convergence is implied by the exis tence of~t o f finite mass and exponen t 6.

18-848283 Acta Mathernatica 153. Imprim6 le 14 D~cembre 1984

266 D. SULLIVAN

Fig. 6

vertically. (The h factors contribute nothing to the exponent.) Since 6> I/2 in the rank

onecusp case (and 6>1 in the rank 2 cusp case) we have 6 > 1 - 6 (and 6 > 2 - 6 in the

rank 2 cusp case). This contradiction proves:

PROPOSITION I. The measure/~ constructed above has no atoms along orbits o f

parabolic cusps.

Acknowledgement. I learned this rather nifty argument from Patterson who em-

ployed it in the Fuchsian case [P].

w 3. The support of ~u and divergence of the Poincar6 series at 6

For our geometrically finite group there are finitely many horospheres whose orbits

under F are disjoint and which rest on all the parabolic points (Fig. 6).

This follows from the Margulis decomposition of any hyperbolic manifold into

thick and thin parts ([T] p. 5.55). If we form the hyperbolic convex hull of the limit set

A(F), remove these horoballs, and divide by F the result is compact ([T] p. 8.20).

It is clear then that any geodesic starting in the convex hull and heading towards a

non-parabolic limit point leaves the horoballs infinitely often. Thus it reenters the

compact region in the quotient infinitely often.

This shows the complement of the parabolic orbits in A(F) consists of radial limit

points ([$2] w 5) (a result of Beardon and Maskit who call these points of approxima-

tion).

By w 1/z gives full measure to the radial limit set. The elementary Corollary 20 of

[$2] w 5 then shows:

PROPOSITION 2. The Poincar~ series gs(x, y) diverges at s=6.

ENTROPY, HAUSDORFF MEASURES AND KLEINIAN GROUPS

�9 § t t

af ae § ~

Fig. 7

267

Remark. This divergence was achieved by Patterson [P] in the Fuchsian case using

a spectral analysis of the Laplacian. He speculated on the likelihood it could be

obtained by more elementary methods.

w The uniqueness of/~ and the ergodicity of/*,/~x/t, and dm~,

Now consider the associated invariant measure for the geodesic flow dm u mentioned in

the introduction. By w 3 this measure is recurrent and so it is ergodic (see [$2] w 4 and

w 5 for these points).

The ergodicity of dmu is equivalent to the ergodicity of F acting on/~x/~. The

ergodicity of/~x/t, implies the ergodicity of F acting on/~.

These ergodicity statements are valid for any finite measure v which satisfies

y*v=ly'l~v and which has no atoms along parabolic cusps. But there can be no atoms

for any v because of the divergence established in w 2.

To see this, consider the orbit of a point x on the horosphere of w 3 (Fig. 7).

Because the convex hull of A(F) less the interiors of horoballs rood F is compact

this orbit is a bounded hyperbolic distance from any point in the convex hull outside the

horoballs.

In particular in the sum for the Poincar6 series for go(x, y) where x belongs to the

convex hull the Z (or Z+Z) terms in the orbit o fx lying on one horosphere add up to a

quantity commensurable with the diameter of the horosphere raised to the 6 power.

PROPOSITION 3. ~ (diameters o f horosphere) ~= oo where the sum is taken over the

orbit of any cusp. (Diameter in metric on unit ball model.)

This proves the v above (which is finite) has no atoms along the cusp orbits. Thus

268 D. SULLIVAN

dm~, vxv , and v are ergodic. This implies v=p because m=~+v)/2 is also ergodic (for

the same reason) and the ratio of p or v to m is invariant.

So we have proved,

THEOREM 1'. There is one and only one probability measure I~ on A(F) satisfying

for yEF, Moreover px/~ is ergodic for the action o f f on A(F)XA(F).

w 5. The eigenfunction q~t,

If the hyperbolic isometry O "-1 translates our fixed point x to a variable point p, the

linear distortion [o'[ on the sphere (with metric coming from rays emanating from x)

only depends on p. Define a function q~, at p by

= fphoro [~ G,(P)

If 2=6(6-2) , q~, is an eigenfunction of the Laplacian with eigenvalue 2. (See [P] and

[$2] w 7.)

By definition q~, is F invariant. Thus by compactness and the continuity of q~, the

size of tp~, is controlled on the convex hull of A(F) mod F outside the cuspidal ends.

If we reconsider the calculation of w l, with the cusp in question put at infinity and

ignoring the factors h (distance) which are not needed now after w 2, we see an estimate

for q~(p) in the cusp

[z ~-6 rank one cusp q0~(p) X [z2_ 6 rank two cusp '

where z is the height of p above x. (And ~ means the ratio is bounded by two

constants.)

These estimates may be viewed in two ways. First, by definition q~, is the p

convex combination of basic ;t-eigenfunctions (the horospherical functions) normalized

to be 1 at x. The basic eigenfunctions behave like the functions c exp ( - 6 . (hyperbolic

distance (p, y))) where y is near infinity (c is exp (6. (hyperbolic distance (x, y)))). Thus a

calculation analogous to that of w 1 consisting of breaking/~ up over the action of T leads

to the above estimate.

Second, and more literally our unique p is constructed from orbit Dirac mass

approximations. Using these approximations the calculation of w 1 becomes the estima-

tion of q~,.

ENTROPY, HAUSDORFF MEASURES AND K L E I N I A N GROUPS 269

Using the estimate on tp~, we can prove

PROPOSITION 4. tp,, is square summable on a unit neighborhood of the conuex hull

of A(F) mod F.

There is no problem on the compact part. The cuspidal ends have cross-sectional

areas decreasing like exp ( - (r . hyperbolic distance)) where r is the rank of the cusp.

(See figure 3 and [T].) The growth of cpF, is by the estimate of this section no more than

exp ((r-0). hyperbolic distance). We use Harnack for positive eigenfunctions to extend

the estimate to a unit neighborhood. We recall O>r/2 from w 2.

Since 2(1-8)<1 if 8>1/2 and 2(2-8)<2 if 8>1, tp 2 is summable in each type of

cuspidal end.

Remark. We don't use it here, but if l<0, tp~, is also square summable outside the

convex hull. In other words,

PROPOSITION. / f F is a geometrically finite group, and if 1<8, q~, defines a

positioe square-summable eigenfunction on H3/F for the eigenvalue 0(0-2).

Proof. If 8>1, qg~, is actually decreasing in a rank one cuspidal end of the convex

hull modF. Then qo~, belongs to L 2 on the a of the unit neighborhood (the only

noncompact part being the rank one cuspidal end).

We may assume the boundary of the neighborhood lifted to H 3 is convex, smooth,

and has bounded curvature. Consider the projection of points outside the neighborhood

to the boundary. The points on a surface at distance R are compressed together by a

factor e -R. So the area element is larger by a factor of e 2R. The value of qo~, has

decreased by a factor of e -hR. Thus the integral over the outside is a double integral

q~2 = dtr dR =1 -~o

where o is the boundary of the unit neighborhood and OR is the surface at distance R.

The integral is finite since 8>1, it being understood the integral is only taken over a

fundamental domain for the action of F.

w 6. The finiteness of dm~, and proof that D= 6(F)

The invariant measure for the geodesic flow dm~ is by definition ([$2])just the arc

length along all geodesics summed via the m e a s u r e fl• 2~ on their endpoints.

270 D. SULLIVAN

Similarly since q0~, is the /, sum for ~ in the sphere at infinity of the basic

horospherical eigenfunctions q0~, q02 at a point is a/ ,x/~ sum of products tpr This

product is invariant under translations along the geodesic connecting ~ and r/and its

value on the geodesic is commensurable to 1/l~-~l 2~ since each cpr is 1 at the center of

the ball.

If we smear out the arc length mass along each geodesic uniformly on its unit

tubular neighborhood and add these according to/~x/~/[~-r/[ 2~ we obtain a smooth

measure whose density function is dominated by a constant, q~2. Thus the total mass of

dm~, in the quotient by F is dominated by the j'q02 on a unit neighborhood of the

support of the set of geodesics both of whose endpoints lie in A(F). In particular we

have for any group F,

PROPOSITION 5. The total mass o f dm~, is dominated by SN92dy where dy is the

Riemann measure and N is a unit neighborhood o f (the convex hull o f A(F))/F.

Remark. When 6>1, we can smear out the geodesics using a function like the

product q0~.9~ restricted to an orthogonal plane (and normalized to have total integral

1). Then the above argument gives for any group

PROPOSITION. l f c~>l, then dma has finite total mass iff q~a is square integrable.

In fact the total mass o f dm l, equals the L 2 norm o f q~,.

Remark. This smearing and comparison argument was supplied by Bill Thurston,

on request.

Using w 5 we have the

COROLLARY. For a geometrically finite group the invariant measure dm~, for the

geodesic f low has finite total mass.

Then using [$2], Theorem 25 we have,

COROLLARY. For a geometrically finite group the Hausdorff dimension of A(F) is

the critical exponent 6(F).

Actually, quoting Theorem 25 [$2] gives the Hausdorff dimension of the radial

limit set equals 6(F). Since (w 2) we only add a countable number of points to get A(F)

the Hausdorff dimension doesn't change.

For the reader who desires a more self-contained argument we note that in the

discussion below of the density function of/~ the argument for Theorem 25 is recapitu-

lated.

ENTROPY, HAUSDORFF MEASURES AND KLEINIAN GROUPS 271

w 7. The density function of p

A metrical study of a measure # involves the density function #(~, r)=/~(ball of radius r

and center ~). The pairs (~, r) can be labeled by polar coordinates from the center of the

ball. We think of (~, r) as corresponding to the endpoint v(~, r) of a geodesic v from the

center x pointing towards ~ of hyperbolic length t=log I/r.

THEOREM. The density function lz(~, r) satisfies

~(~, r) ~, r~cp~(v(~, r))

where ~ means the ratio is bounded above and below, and ~ belongs to A(F).

Proof. The inequality /x(~, r)~<constant r~q~,(v(~, r)) is true for all groups. This

follows from the formula w ~0~(p)=S [o'l~d/~, and the fact that [o'[~l/r on a ball of

radius r about ~.

If the other inequality were not true we would have a sequence (~i, ri) so that the

ratios l~(~i, ri)/r~i ~,(v(~i, ri))-*O.

Change the base point of the group to v(~i, ri) and form a geometric limit group,

(IT] p. 9.1). The measures [a~[~-/~ where oF 1 takes the center to v(~i, ri) have total

mass ~,(v(~,., r~)) by definition. Thus dividing by the total mass we can form a limit

measure v which will satisfy y*v=lT'[% for each y in the limit group.

Let us suppose ~i-->~ and ~* is antipodal to ~. Since [oi[r~I/ri on a disk of size k. 1/r centered at ~ on the sphere for any k we conclude that v is concentrated at the antipodal

point ~*. It follows that the limit group is a subgroup of the parabolic group associated

to ~*.

Since the ~i belong to A(F) so does ~. For a geometrically finite group one can

check that one of these limit groups is either not entirely parabolic or contains

nontrivial elements fixing ~.

That is either a subsequence v(~,., r~) stays in the compact part of the (convex

hull)/F or it goes out a cuspidal end. The limits in these two cases differ from a

parabolic group at ~*. This proves the theorem.

w 8. The packing measures

If U is an open set in a metric space A and ~p(r) is a positive function define the packing

measure vp (relative to ~0) by

vp(U) = lim sup E ~P (ri) e---~O P( t ) i

272 D SULLIVAN

where P(e) is the collection of packings of U with radii ~<e. (A packing of U is a

collection B1, B2 ... . of metric balls in U of radii rl,/'2 . . . . so that the sum of two radii is

less than the distance between centers.)

The outer measure determined by the values vp(U), vp(X)=infx=vvp(U), defines

a countably additive Borel measure because vv(XtJ Y)=vp(X)+vp(Y) if X and Y are

separated by a positive distance [R].

Now we produce finite positive measures equivalent to packing measures.

Suppose p is a finite positive measure with compact support in Euclidean n-space

say. Define the density function of/~, kt(x, r), by

/t(x, r) =p(ball of radius r, center x).

PROPOSITION. There is a set A o f full t~ measure so that p is equivalent to the ~p(r)

packing measure o f A i f there are constants c, C so that

c < ~ p(x,r) <. C ~(r) i.o.

for almost all x and r<~ro. (~<i.o. means that the inequality holds for a sequence o f

ri---~O, depending on x.)

Proof. If B1,Bz .... is a packing of Uc{x: inequalities hold)=A by balls of radii

r~, r2, ...<~e<~ro then

E lP(ri )<~l E It(Bi)<~lkt(U)"

Thus the ~0(r) packing measure Vp is a finite measure and for every Borel set

vp(X)<_( I/c) /~(X).

For each x in A, there is a sequence of balls B; with radii r~---~0 so that

#(x, ri)<~CW(r). By the covering lemma ([F], Theorem 2.8.14) there are arbitrarily fine

coverings of any X c A by balls centered on X and falling into k(n) collections of disjoint

balls. One of these disjoint collections must contain 1/k(n).p(X) of the mass of/~. Thus

for this packing r, ~p(ri)~(1/C" k(n))~(X).

For any fixed neighborhood U of X sufficiently fine such coverings define packings

contained in U. Thus vp(U)>-(1/C.k(n))kt(X)for any U c X . So vp(X)>-(1/C'k(n))l~(X).

Thus Vp and/t are equivalent and the Radon-Nikodym derivative dlddvp lies in the

interval [c, C" k(n)].

Remark. If one replaces the hypothesis on p in the proposition by

ENTROPY, HAUSDORFF MEASURES AND KLEINIAN GROUPS 273

c <. ~t(x, r) - - ~< C , r ~< r 0, /~ a . a . x ,

i .o. ~p(r)

then/~ is equivalent to the covering Hausdorff measure relative to the function ~p(r).

This fact is well-known [F] and the proof is similar to the one above.

COROLLARY (of proof). I f l~ is a probability measure satisfying

liminf/~(~, r) is finite and positive, r-*O ~)(r)

there is an increasing sequence o f subsets Ancsupp/~ so that/~ is equivalent to a weak

limit o f the packing measures vn o f A~.

Proof. Let A~ be the set where the liminf belongs to [l/n,n], and the lower

inequality is true for r << - l/n. Then let A .cA~ be such that ~(B n An)//~(B)> 1/2 for balls B

centered on An of radius <rn. By the above proposition/~/A~ is equivalent to the

packing measure vn of A n. By the density theorem [F] 2.9.11 it follows easily that

Vn-'->/~-

w 9. Proof of Theorem 2

Let ~ be the canonical geometric measure on the limit set of a geometrically finite

Kleinian group. The metrically defined density function of/~,/~(~, r), is controlled by

the theorem of w 7 which says

/~(~, r) ~ rrq~,(v(~, r)), r <<- ro.

Now suppose all cusps have rank I>6 (=D). Then by the estimates of w 5, %, is

bounded from below. By ergodicity of the geodesic flow w 6 almost all geodesics enter

the compact part of the convex hull (mod F) infinitely often. Thus there are constants

c, C so that

c ~</~(~' r) ~ C. rt~ i.o.

Thus by w 8/~ is equivalent to the packing measure associated to r ~. In particular

the packing measure Vp is finite and positive. Since a finite packing measure for r ~

clearly satisfies

y*v=[y'[%, y E F

we have ~=vp by uniqueness of such measures.

2 7 4 D. SULLIVAN

se lmlt

Fig. 8

Note that if some cusp has rank >d, the tp, is unbounded and the ratio/z(~, r)/r ~ is

unbounded from above for almost all ~ (again using ergodicity of the geodesic flow).

Applying the covering lemma as in the proof of the proposition in w 8 shows the

Hausdorff covering measure is zero. Thus Vp~=Vc.

Similarly if all cusps have rank ~<d we arrive at the dual inequalities

c ~< /z(~, r___~) ~< C i.o. r ~

and/z equals the Hausdorff covering measure relative to r ~. Also if some cusp has rank

<d, then the ratio/~(~, r)/r ~ is not bounded from below. Using the covering lemma

shows Vp= o%

This proves most of the entries in the table following Theorem 2.

For the rest we recall Beardon's result (indicated in w 3) that c5>k/2 (k any rank of a

cusp), and note that if D=2 the Poincar6 series diverges at s--2 so F cannot have any

points of discontinuity and thus there are no rank one cusps.

w 10. Counting orbits of the geodesic flow: entropy

Let us estimate the maximal number N(e, t, B) orbits of the geodesic flow which start in

a ball B and are e apart some time before time t (Fig. 8). (We only consider geodesics

both of whose endpoints are in the limit set A(F).)

For each ~ in A(F) define a set U~ as indicated in the figure 8 (a).

Figure 8 (a) shows we need to consider rays emanating from B with endpoints in

U~nA(F). Then a finite number of these U~'s will cover. Now if two geodesics are e-

part

ENTROPY, HAUSDORFF MEASURES AND KLEINIAN GROUPS

Fig. 9

275

apart in H3/[" s o m e time before time t, their lifts to H 3 will also be e-apart some time

before time t. Conversely, if two geodesics starting from nearly the same point in (a lift

of) B are e apart at some time s<t they project to geodesics which at time s are either e-

apart in H3/F or which lie in the e-thin part of H3/F. (Because outside the e-thin part the

covering project ion is an isometry on e-balls.) (Fig. 9.)

Le t us ignore this latter complication for the moment. If so, we end up counting the

number of points in the limit set intersect U no two of which are closer than e e - t .

We can use the canonical geometr ic measure to estimate this number. For exam-

ple, if there are no cusps/~(~, r)~r ~ since ~p~, is bounded above and below w 5. Then

there are no more than r~(2/e)~e ot such points�9 Similarly, in a cover by e balls there

must be at least r~(1/e)Oe ~t such balls. If we take the log of this number divide by t ,

take the l imsup as t---~o~ and then the limit as e ~ 0 we get Bowen's definition of the

topological en t ropy of the geodesic flow (restricted to those orbits with both endpoints

in the limit set). Thus we have (since the complication about t-thin parts is not relevant

if there are no cusps),

THEOREM. For a geometrically finite group without cusps, the topological entro-

py(l ) of the geodesic flow (on the compact set o f geodesics with endpoints in the limit

set) is the Hausdorff dimension o f the limit set A(F).

It is natural to ask what is the measure theoretic entropy relative to the finite

invariant measure drag const ructed f rom the canonical geometric measure/~. We can

see the answer is still the Hausdorf f dimension (even in case there are cusps). In

particular, dm~, above is a measure which maximizes the entropy.

(~) Calculated using the natural logarithm.

276 D. SULLIVAN

Let us divide the (convex hull)/F minus neighborhoods of cusps (the e-thin part of

H3/F) into finitely many cells of small diameter. By ergodicity relative to dm,, a

geodesic only spends a small fraction f of its time in the thin part or near the walls of

these cells (the shaded part of the figure).

Thus if we only distinguish orbits which are in different cells for more than a

fraction 2 fo f the time up to time t we are again counting metrically separated orbits in

H 3 starting near one point of a lift B. The calculation of the exponential growth rate of

the number goes as before with two further considerations.

(i) We are looking for e e - r separated points again but t' lies in [ (1-2f) t , t]. Since

f is arbitrarily small this complication won't matter for the growth rate.

(ii) Now the/z mass of a relevant ball of radius e -c is e-t'~q~/,(v(~, t')). The factor

q~, however has arbitrarily small exponential effect for most geodesics because dm,

almost all geodesics are within o(t) of the compact part ([$2] Corollary 19).

We find the exponential growth rate of the number of orbits counted using these

partitions and looking most of the time works out to be 5. According to Feldman and

[BK] (as explained to me by Dan Rudolph) these rates (which in our case are constantly

5) converge to the measure theoretic entropy. This proves the

THEOREM. For a geometrically finite group the measure theoretic entropy(2) of the geodesic f low relative to the canonical measure dm~, is the Hausdorff dimension o f

the limit set.

Remark. (Pesin theory of exponents.) For the geodesic flow there is a familiar

Hopf-Anosov picture of expanding and contracting foliations transverse to the flow.

There is a uniform expansion (contraction) in two senses: for a time t flow at each

point, each expanded (contracted) vector is expanded (contracted) by el(e-t). The

measure dm~ conditioned to the leaves of the expanding (contracting) foliation is by

construction equivalent to the geometric measure/~ of dimension 6. It is expanded

(contracted) uniformly by e~l(e-6t). Thus if one imagines Pesin theory of exponents

and the Pesin-Margulis entropy formula being valid in a fractal dimension 6, the above

result is consistent. Note that when H3/F has finite volume, the answer 2 for the

entropy does follow formally from the Pesin-Margulis formula.

(2) Calculated using the natural logarithm.

ENTROPY, HAUSDORFF MEASURES AND KLEINIAN GROUPS 277

References

[B] BOWEN, R., Hausdorff dimension of quasi-circles. Publ. Math. I.H.E.S., 50 (1979), 11-26. [B2] BEARDON, A. F., The Hausdorff dimension of singular sets of properly discontinuous

groups. Amer. J. Math., 88 (1966), 722-736. [BK] BRIN, M. & KAaOK, A., On local entropy. Geometric Dynamics, Proceedings in Rio de

Janeiro 1981. Lecture Notes in Mathematics, 1007, pp. 30-38. [F] FEDERER, H., Geometric measure theory. Springer Grundlehren, Series Band 153. [P] PATTERSON, S. J., The limit set of a Fuchsian group. Acta Math., 136 (1976), 241-273. [R] ROGERS, C. A., Hausdorffmeasures. Cambridge University Press 1970, chapter 1. [S1] SULLIVAN, D., Disjoint spheres, diophantine approximation, and the logarithm law for

geodesics. Acta Math., 149 (1983), 215-237. [$2] - - The density at infinity of a discrete group of hyperbolic isometries. Publ. Math.

I.H.E.S., 50 (1979), 171-209. IT] THURSTON, B., Geometry and topology of 3-manifolds. Notes from Princeton University,

1978.

Received March 30, 1981